Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Existence of positive solutions to discrete secondorder boundary value problems with indefinite weight | Gao et al. Advances in Difference Equations 2012 2012 3 http content 2012 1 3 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Existence of positive solutions to discrete second-order boundary value problems with indefinite weight Chenghua Gao Guowei Dai and Ruyun Ma Correspondence gaokuguo@163. com Department of Mathematics Northwest Normal University Lanzhou 730070 P. R. China Springer Abstract Let T 1 be an integer T 1 2 . T . This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problems A1 2u t - 1 rm t f u t 0 t e T u 0 u T 1 0 where r 0 is a parameter m T R changes its sign m t 0 for t e T and f R R is continuous. Also we obtain the existence of two principal eigenvalues of the corresponding linear eigenvalue problems. MSC 2010 39A12 34B18. Keywords discrete indefinite weighted problems positive solutions principal eigenvalue bifurcation existence 1 Introduction Let T 1 be an integer T 1 2 . T . This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problem BVP A2u t - 1 rm t f u t 0 t e T u 0 u T 1 0 where r 0 is a parameter f R R is continuous m t 0 for t e T and m T R changes its sign . there exists a proper subset T of T such that m t 0 for t e T and m t 0 for t e T T . BVPs with indefinite weight arise from a selection-migration model in population genetics see Fleming 1 . That an allele A1 holds an advantage over a rival allele A2 at some points and holds an disadvantage over A2 at some other points can be presented by changing signs of m. The parameter r corresponds to the reciprocal of the diffusion. The existence and multiplicity of positive solutions of BVPs for second-order differential equations with indefinite weight has been studied by many authors see for example 2-5 and the references therein. In 2 using .
